Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #63091 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C2×#63093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C4 | C27 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
55980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 66 points:
(1,2);(3,4,9,14)(5,15,19,36)(6,11,21,32)(7,12,23,34)(8,25,27,13)(10,26,30,18)(16,37,40,57)(17,38,42,58)(20,31,44,55)(22,47,46,33)(24,50,49,35)(28,48,53,41)(29,51,54,43)(39,59,60,66)(45,63,62,56)(52,64,65,61);(3,5,9,19)(4,10,14,30)(6,16,21,40)(7,17,23,42)(8,26,27,18)(11,28,32,53)(12,29,34,54)(13,36,25,15)(20,39,44,60)(22,48,46,41)(24,51,49,43)(31,52,55,65)(33,57,47,37)(35,58,50,38)(45,64,62,61)(56,66,63,59);(3,6)(4,11)(5,16)(7,20)(8,22)(9,21)(10,28)(12,31)(13,33)(14,32)(15,37)(17,39)(18,41)(19,40)(23,44)(24,45)(25,47)(26,48)(27,46)(29,52)(30,53)(34,55)(35,56)(36,57)(38,59)(42,60)(43,61)(49,62)(50,63)(51,64)(54,65)(58,66);(3,7)(4,12)(5,17)(6,20)(8,24)(9,23)(10,29)(11,31)(13,35)(14,34)(15,38)(16,39)(18,43)(19,42)(21,44)(22,45)(25,50)(26,51)(27,49)(28,52)(30,54)(32,55)(33,56)(36,58)(37,59)(40,60)(41,61)(46,62)(47,63)(48,64)(53,65)(57,66);(7,23)(12,34)(17,42)(20,44)(24,49)(29,54)(31,55)(35,50)(38,58)(39,60)(43,51)(45,62)(52,65)(56,63)(59,66)(61,64);(6,21)(11,32)(16,40)(20,44)(22,46)(28,53)(31,55)(33,47)(37,57)(39,60)(41,48)(45,62)(52,65)(56,63)(59,66)(61,64)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494194);